Navigate to your Library tab and click once to bring up the Drop-Down tab. Open All Games, and you might see that your Purchased Game is listed, but greyed out. Click to select, and it will give you the option to Install.

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Method 2.

On the main Steam Page, highlight your Profile next to Community (it says "New" directly above your username.

At the bottom of the drop-down list, is Inventory. Click it.

When Inventory opens, make sure that you select Steam (there might be other game inventories listed).

See if GTA is listed there, and if it is, try to Click / Select it and it will ask if you would like to install it / move it to your library (to enable you to install it).

Now go to Library and you will see the game there. When you Highlight it, it will allow you to Play / Install. Click and it will begin downloading / updating.